Chapter 2 Projections and Unitary elements
The K-theory of a
In this chapter we derive the facts needed about projections and unitary elements with emphasis on the equivalence relation defined by homotopy and also — for projections — Murray-von Neumann equivalence and unitary equivalence.
Let
Definition 2.1.2.
Let
Products of homotopic unitaries
If
Indeed, find continuous paths
Lemma 2.1.3
Let
Proof of 1:
It follows from the 2.1.13 Theorem (Continuous functional calculus) That if
This implies that
For each
Since
Since the map is continuous for t at every x, in particular it should be continuous at just h
Therefore
Proof of 2:
If
Then
Set
The main idea here is that if the element has as spectrum only part of the unit circle, we can define a branch on the argument function acting on the spectrum of the element, from there we can access our element as an argument of some self adjoint element on the unit circle, then we apply part 1 to this exponential.
Proof of 3:
If
This in turn implies that
This proof exploits the relationship between the spectral radius of a normal element and its norm, along with the convenient fact that
Each unitary in
Corollary 2.1.4
The unitary group in
Lemma 2.1.5 (Whitehead)
Let
In
It follows in particular that
Proof:
From Lemma 2.1.3 we get that
Proposition 2.1.6.
Let
Let
It follows from 1. and Lemma 2.1.3 (1), that
Since
Take
The complement
In conclusion,
Therefore 2. and 3. hold.
Lemma 2.1.7.
Let
Proof of 1:
A unital
Conversely, if
Then
set
Then
Reflection:
The first containment follows from continuity of the map.
The second relies on surjectivity, we have this decomposition of elements in
Proof 2:
Follows immediately from 1. and Lemma 2.1.5 (Whitehead).
[Insert detailed rumination]
Proof 3:
If
Let
The group of unitary elements in
Part 2. of the following proposition says that
A subspace
For each element
Proposition 2.1.8
Let
Multiplication in a
The former of these maps is continuous because the involution and multiplication is continuous. It now suffices to show that
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Aside:
Lemma 1.2.5. Says that any continuous function
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If
Let
Then
For each
Thus
Proof of 3: If
Theorem 2.1.9. (The polar decomposition)
The factorization
Theorem 2.1.10 (Carl Neumann Series)
Let
Let
Let
Definition 2.2.1
Recall that an element
We can use this to show that the Murray-von Neumann relation is transitive.
Proposition 2.2.2
Let
It follows that
Proposition 2.2.4
If
Proof:
Put
Secondly,
each
Now invoke Urysohn’s Lemma to define
Then
The path
The next proposition says that if two self-adjoint elements in a unital C* algebra are similar, then they are unitarily equivalent.
Proposition 2.2.5
Let
Proof:
The equation
Hence
Proposition 2.2.6
Let
Proof:
Throughout this proof,
Suppose first that
Then, since
(It is clear that for each t, we have a projection, and the path is the desired one, continuity in P(A) requires A to be an ideal I.E, conjugation will stay in A)
Conversely, if
It therefore suffices to prove the implication in the case where
Put
Let
Then
Note:
We have three equivalence relations
That
Homotopy
Proposition 2.2.8 says that the three relations are actually the same when we pass to matrix algebras.
Proposition 2.2.7
Let
Proposition 2.2.8
Let
2.2.10 Liftings
Suppose that
Proof of 2 (lifting self-adjoints):
Let
to arrange that the lift has the same norm as
Proof of 1 (lifting to elements with same norm):
Let
By (1.4)
Proof of 3 (positive elements do lift):
Let
Proof of 4
See exercise 9.4(2)
Proof of 5 (Projections do not lift):
Let
Proof of 6 (Unitaries don’t lift in general):
Exercise.
Definition 2.3.1 (The semigroup of projections of A)
Put
Define the relation
Suppose that
Define a binary operation
so that
The relation
Proposition 2.3.2.
Let
Definition 2.3.3. (The semigroup of projections)
With
It follows from Proposition 2.3.2. that this operation is well-defined and that
In the next chapter we will construct for each unital C*-algebra